Equivalent form of Rational Numbers
We will learn how to find the equivalent form of rational numbers expressing a given rational number in different forms and the equivalent form of the rational numbers having a common denominator.
1. Express \(\frac{-54}{90}\) as a rational number with denominator 5.
Solution:
In order to express \(\frac{-54}{90}\) as a rational number with denominator 5, we first find a number which gives 5 when 90 is divided by it.
Clearly, such a number = (90 ÷ 5) = 18
Dividing the numerator and denominator of \(\frac{-54}{90}\) by 18, we have
\(\frac{-54}{90}\) = \(\frac{(-54) ÷ 18}{90 ÷ 18}\) = \(\frac{-3}{5}\)
Hence, expressing \(\frac{-54}{90}\) as a rational number with denominator 5 is \(\frac{-3}{5}\).
2.
Fill
in the blanks with the
appropriate number in the numerator: \(\frac{5}{-7}\) = \(\frac{.....}{35}\) = \(\frac{.....}{-77}\).
Solution:
We have, 35 ÷ (-7) = - 5
Therefore, \(\frac{5}{-7}\) = \(\frac{5 × (-5)}{(-7) × (- 5)}\) = \(\frac{-25}{35}\)
Similarly,
we have (-77) ÷ (-7) = 11
Therefore, \(\frac{5}{-7}\) = \(\frac{5 × 11}{(-7) × 11}\) = \(\frac{55}{-77}\)
Hence, \(\frac{5}{-7}\) = \(\frac{-25}{35}\) = \(\frac{55}{-77}\)
More examples on equivalent form of rational numbers:
3. Find an equivalent
form of the rational numbers \(\frac{2}{9}\) and \(\frac{5}{6}\) having a common denominator.
Solution:
We have to convert \(\frac{2}{9}\) and \(\frac{5}{6}\) into equivalent rational numbers having common denominator.
Clearly, such a denominator is the LCM of 9 and 6.
We have, 9 = 3 × 3 and 6 = 2 × 3
Therefore, LCM of 9 and 6 is 2 × 3 × 3 = 18
Now, 18 ÷ 9 = 2 and 18 ÷ 6 = 3
Therefore, \(\frac{2}{9}\) = \(\frac{2 × 2}{9 × 2}\) = \(\frac{4}{18}\) and \(\frac{5}{6}\) = \(\frac{5 × 3}{6 × 3}\) = \(\frac{15}{18}\).
Hence, the given rational numbers with common denominator are \(\frac{4}{18}\) and \(\frac{15}{18}\).
4. Find an equivalent form of the rational numbers \(\frac{3}{4}\), \(\frac{7}{6}\) and \(\frac{11}{12}\) having a common denominator.
Solution:
We have to convert \(\frac{3}{4}\), \(\frac{7}{6}\) and \(\frac{11}{12}\) into equivalent rational numbers having common denominator.
Clearly, such a denominator is the LCM of 4, 6 and 12.
We have, 4 = 2 × 2, 6 = 2 × 3 and 12 = 2 × 2 × 3
Therefore, LCM of 4, 6 and 12 is 2 × 2 × 3 = 12
Now, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1
Therefore, \(\frac{3}{4}\) = \(\frac{3 × 3}{4 × 3}\) = \(\frac{9}{12}\), \(\frac{7}{6}\) = \(\frac{7 × 2}{6 × 2}\) = \(\frac{12}{12}\) and \(\frac{11}{12}\) = \(\frac{11 × 1}{12 × 1}\) = \(\frac{11}{12}\)
Hence, the given rational numbers with common denominator are \(\frac{9}{12}\), \(\frac{14}{12}\) and \(\frac{11}{12}\).
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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